In Section 2 we recall the definition and atomic decomposition of the Hardy
spaces with variable exponents given in [1]}. We also state three
crucial lemmas, two of them refering to estimations of the \(L^{p(.)}\left(\mathbb {R}^{n}\right) \)  norm of the characteristic functions of cubes and
the other one about the vector valued boundedness of the fractional maximal
operator.
